Instructions

Try your best to answer the questions above. Type your answers into
the boxes provided leaving no spaces. As you work through the exercise regularly click
the "check" button. If you have any wrong answers, do your best to
do corrections but if there is anything you don't understand, please
ask your teacher for help.

When you have
got all of the questions correct you may want to print out this page
and paste it into your exercise book. If you keep your work in an
ePortfolio you could take a screen shot of your answers and paste
that into your Maths file.

Transum.org

This web site
contains over a thousand free mathematical activities for teachers and
pupils. Click here to go to the
main page which links to all of the resources available.

Comment recorded on the 5 April 'Starter of the Day' page by Mr Stoner, St George's College of Technology:

"This resource has made a great deal of difference to the standard of starters for all of our lessons. Thank you for being so creative and imaginative."

Featured Activity

Roman Numerals Quiz

You may understand our number system better by learning about another number system. A basic knowledge of Roman numerals will allow you to complete level one of this self marking quiz. Beyond level one will require a little more!

Answers

There are answers to this exercise but they are available in this space to teachers, tutors and parents who have logged in to their Transum subscription on this computer.

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Go Maths

Learning and
understanding Mathematics, at every level, requires learner
engagement. Mathematics is not a spectator sport. Sometimes
traditional teaching fails to actively involve students. One way to
address the problem is through the use of interactive activities and
this web site provides many of those.
The Go Maths page is
an alphabetical list of free activities designed for students in Secondary/High school.

Maths Map

Are you looking for something specific? An exercise to supplement the topic you are studying at school at the moment perhaps. Navigate using our Maths Map to find exercises, puzzles and Maths lesson starters grouped by topic.

Teachers

If you found this activity useful don't forget to record it in your scheme of work or learning management system. The short URL, ready to be copied and pasted, is as follows:

Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world.
Click here to enter your comments.

Help

A decimal with a repeating digit (or set of digits) is called a recurring decimal.

For example \(0.77777777...\) is a recurring decimal and is called "nought point seven recurring"

\(9.247347347...\) is also a recurring decimal and is called "nine point two four seven recurring"

The period of a recurring decimal is the number of digits in the repeating section so for the second example above the period is three.

A more efficient way of writing out a recurring decimal is by only writing the repeating digit once but putting a dot over the first and last number in the repeating sequence. Another method is drawing a line over the repeating digit or digits.Here are some examples.

\(0.333333333... = 0.\dot 3 = 0.\overline 3\)

\(0.76531531531... = 0.76\dot 53\dot 1 = 0.76\overline{531}\)

A fraction can be converted to a decimal using long division; dividing the numerator by the denominator. If the decimal is recurring the repeating pattern of numbers will be spotted in the long division working. The following example shows the repeating patterns when converting \( \frac{7}{11} \) to a decimal:

There are two common methods for converting a recurring decimal to a fraction:

Method 1

1 repeating digit

Let the recurring decimal be represented by \(x\)

$$x = 0.8888888...$$

Multiply both sides by 10 (as there is one repeating digit)

$$10x = 8.8888888...$$

Subtract the first equation from the second

$$9x = 8$$
$$x = \frac{8}{9}$$

2 repeating digits

Let the recurring decimal be represented by \(x\)

$$x = 1.36363636...$$

Multiply both sides by 100 (as there are two repeating digits)

$$100x = 136.36363636...$$

Subtract the first equation from the second

$$99x = 135$$
$$x = \frac{135}{99}$$
$$x = \frac{15}{11}$$

3 repeating digits

The method is the same but multiply both sides by 1000.

Method 2

1 repeating digit

Example: convert \(0.8888888...\) to a fraction.

This method requires you to know that \(\frac19 = 0.1111111...\)

\(0.8888888...\) is exactly eight times \(0.1111111...\)

$$\therefore 0.8888888... = \frac{8}{9}$$

2 repeating digits

Example: convert \(0.45454545\) to a fraction in its lowest terms.

This method requires you to know that \(\frac{1}{99} = 0.01010101...\)

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.